STABILITY OF A FEW CURVED 

PANELS SUBJECTED TO SHEAR 

Original report dated May 1947 

Information Reviewed and Reaffirmed 

August 1962 

No. 1571 

In Cooperation with the University of Wisconsin 
1 

STABILITY OF A FEW CURVED PANELS SUBJECTED TO SHEAR 

By 

EDWARD W. KUENZI, Engineer 

2 Forest Products Laboratory, Forest Service 

U. S. Department of Agriculture 

Summary 

The major portion of the experimental work of this study was on curved plywood plates subjected to shearing loads. One large specimen of sandwich 

construction was tested as a preliminary check on the extension of the theory 

for plywood to the sandwich type of construction. 

The results of the tests on plywood plates indicated that the buckling stress of 

a curved plate is approximately equal to the sum of the buckling stress of a 

complete cylinder subjected to torsion and the buckling stress of a flat plate, 

of the same size as the curved plate, subjected to shear. 

The test on the single plate of sandwich type of construction indicated that the 

theory for plywood can be extended to sandwich constructions. Curves are 

presented showing the theoretical buckling coefficients for sandwich material. 

Introduction 

A sandwich construction comprised of thin facings bonded to a light-weight 

core has several advantages when considered for use in aircraft structures. 

The construction is light in weight, is stiff, and therefore will retain its proper 

shape under load, has a smooth surface that reduces drag, and can be of monocoque design thus eliminating ribs and longerons. 

1

This report is one of a series of progress reports prepared by the Forest 

Products Laboratory. Results here reported are preliminary and may be revised as additional data become available. Report originally issued May 1947. 

2 

Maintained at Madison, Wis., in cooperationwith the University of Wisconsin. 

Rept. No. 1571 
The main objective of this study was to investigate the elastic behavior of 

curved plates of sandwich material subjected to shearing loads such as those 

imposed upon the wings and fuselage of an airplane. 

Methods of testing and of confirmation of design criteria were determined by 

testing 47 curved plates of plywood. A preliminary check on the buckling of 

a curved plate of sandwich construction was obtained by testing one specimen 

consisting of two curved sheets about 4 feet square glued to loading rails as 

shown in figure 1. A more complete investigation, including such variables 

as size, curvature, and different materials, is desirable but was not justifiable at the time. Probably tests on specimens of the same size as those used 

in a structure or tests on the structure itself would best provide the necessary 

information. 

Design Criteria 

In order that buckling stresses of curved plates in shear can be predicted, it 

is necessary to obtain design criteria that will represent the experimental 

data. The study of curved sandwich plates in axial compression, presented 

in Forest Products Laboratory Report No, 1558, shows that the buckling 

stress of a curved plate is equal to the sum of the buckling stress of a complete cylinder and the buckling stress of a flat plate the same size as the 

curved plate. It is logical to apply a similar analysis to a curved plate in 

shear; for example, to assume that the buckling stress of the curved plate in 

shear is equal to the sum of the torsional buckling stress of a complete 

cylinder of the same length and curvature as the curved plate and the shear 

buckling stress of a flat plate of the same size as the curved plate. 

Cylinder in Torsion 

The buckling of plywood cylinders in torsion is discussed in Forest Products 

Laboratory Report No. 1529. The theoretical formula for the buckling stress 

of a thin-walled plywood cylinder, as given in that report, is 

(1) 

where k is a coefficient depending upon the elastic properties and the parameter J. Values of k for plywood of three veneers with the faces half as thick 

as the core are shown in figures 2, 3, 4, and 5 for different ratios of 

where μLT is the modulus of rigidity of the plywood. 

Rept. No. 1571 - 2-
J = 

b = 

h = 

r = 

EL = 

The 

b2 

hr 

length of cylinder 

thickness of plywood 

radius of curvature of cylinder 

modulus of elasticity of the veneer, measured parallel to the direction 

of the grain 

corresponding formula for thin-walled cylinders of sandwich construction is 

(2) 

if the facings and core are assumed to be isotropic (derivation in appendix 1). 

The value of k' depends upon the parameter 

(Values of k' are shown in fig. 6) 

where E1 = modulus of elasticity of the sandwich plate in bending. 

Ea = modulus of elasticity of the sandwich plate in compression. 

Values of E1 and Ea can be computed by the following equations, if the core 

is assumed to contribute no stiffness to the sandwich. 

where Ef = modulus of elasticity of the facing material 

c = core thickness 

h = total thickness of the sandwich 

Flat Plate in Shear 

The elastic behavior of flat plywood plates subjected to shear is discussed in 

Forest Products Laboratory Reports 1316 and 1316-H. The formula for the 

buckling stress is 

(3) 

Rept. No. 1571 -3-
where ks is a coefficient depending upon the elastic properties of the plywood 

and the ratio b/a. Values of ks are determined by the methods given in Forest 

Products Laboratory Reports Nos. 1316 and 1316-H, as shown in appendix 3. 

a = width of flat plate. For curved plates this width was taken to be the circumferential dimension. 

If the flat plate is of isotropic sandwich material, formula (3) becomes 

where ks = ca is obtained from the curve for = 1 of figure 8, Report No. 

1316. Poisson's ratio. = 0.91 (approximately) for 

isotropic materials . 

The results of a few tests on flat plates of sandwich material are given in 

Forest' Products Laboratory Report No. 1560. 

Curved Plate in Shear 

The buckling stress of the curved plate subjected to shear is assumed to be 

equal to 

of Results." 

Confirmation of this assumption will be given in the "Discussion 

Experimental Work 

The curved-plate specimens were constructed as shown in the sketch of figure 

7. Each specimen was made of two curved plates. This arrangement allowed 

the use of the same loading apparatus that was used to test flat plates in shear. 

Materials 

The plywood specimens were made of four plies of yellow birch veneers 1/64 

inch thick. The grain directions of the two center plies were parallel; thus 

the four-ply plywood was equivalent to three-ply plywood having face plies half 

as thick as the core ply. The veneers were bonded together with Tego film 

glue, set in a hot press. The sheets of plywood were made large enough to 

furnish several coupons which were tested to obtain the elastic properties of 

plywood. 

Rept. No. 1571 - 4-
The sandwich specimen comprised 0.0 12-inch 24ST alclad-aluminum facings 

bonded to a 1/8-inch end-grain balsa core. The curved plates of this specimen were bag molded as described in Forest Products Laboratory Report No. 

1574. 

Manufacturing the Specimens 

The curved plywood test specimens, as shown in figure 7, were made by gluing 

1- by 4-inch maple loading strips (B) to the edges of the plywood while the plywood was flat. Then the plywood was bent and glued to maple end blocks (A). 

The maple strips (B) were screwed to a pine filler (F). Finally holes were 

drilled for the loading pins (LP). The inside dimensions of the loading frame 

were 10 by 20 inches. 

The curved plate of sandwich construction was made in a similar manner, 

except that the plates were molded to the proper curvature prior to fitting on 

the loading frame. The sandwich plate was 50 inches wide and 50 inches long, 

curved to a mean radius of 93 inches. 

Bending and tension tests were made on coupons of plywood. No coupons of 

the sandwich construction were tested, 

Testing Procedures 

The curved plywood specimens were tested in a hydraulic testing machine. 

The sandwich specimen was tested in a mechanically operated testing machine. 

The method of test is shown in figure 1. The bottom loading block rested on 

an adjustable spherical base. The upper loading block was fastened to the 

upper head of the testing machine. After the specimen was placed in a vertical position, the spherical base was adjusted so that a good bearing was assured 

on all of the loading rollers. The load was applied at a slow uniform rate until 

buckling occurred. The buckles usually appeared suddenly, with an accompanying decrease in load; but if the plates were nearly flat, they appeared slowly 

and it was necessary to determine the critical load by observation of the lateral 

deflection. The buckling load was picked at the point of greatest curvature of 

the load-deflection curve. The plywood specimens (curved plates and coupons) 

were conditioned and tested in an atmosphere having a relative humidity of 65 

percent and a temperature of 74° F. 

The methods of testing the coupons are given in appendix 2. 

Rept. No. 1571 -5-
Computation of Results 

The test results are shown in table 1. The experimental buckling stress was 

computed by dividing the component of force parallel to the straight edge of 

the specimen by the area of the plates at that edge. Let = the experimental 

buckling stress, then 

where P is the buckling load, ß is the angle between the diagonal and the long 

side of the specimen, b is the long dimension of the specimen, and h is the 

thickness of one curved plate. 

The theoretical buckling stresses were computed by means of the formulas 

given in the preceding section on "Design Criteria." 

The values of k used in this study were obtained by extending the work of 

Forest Products Laboratory Report No. 1529 to include values for J < 250, 

and are given in the curves of figures 2, 3, 4, and 5 for three-ply plywood 

E 1 having a ratio of E1 + E2 

equal to 0.835. Values of k' for use in calculating 

the buckling stress of isotropic sandwich constructions are shown in figure 6. 

This curve of k' vs. J' was obtained by the analysis given in appendix 1. 

The values of ks were obtained from the curves of Forest Products Laboratory Reports Nos. 1316-H and 1316. 

Sample computations for plywood and for sandwich construction are given in 

appendix 3. 

Presentation and Discussion of Results 

Experimental and computed values of the buckling stresses are given in table 

1 and comparisons of them are shown in figures 8 and 9. Figure 8 shows the 

stress values compared with the computed buckling stresses 

of a complete cylinder in torsion. The experimental values are as much 

as 70 percent higher, provided each plate is not half of a completed cylinder, 

experimental 

for example, In figure 9 the experimental buckling stress is plotted 

against The points fit the theory much better than if just is 

Rept. No. 1571 -6-
used for a computed value. The experimental values for specimens that are 

complete cylinders are lower than computed values, 

No attempt has been made to analyze the distribution of stresses along the 

curved edge of the plate. The difference between experimental and computed 

values for specimens of a complete cylinder indicates that the plates are probably not subjected to pure shear. 

The experimental buckling stress of the one specimen of sandwich construction agreed well with the theoretical value of 

specimen 

The comparison can 

be seen by referring to figures 8 and 9. The was large and the effect 

of transverse shearing deformations was small. Since the core was relatively 

weak and easily deformed, transverse shearing deformations might have been 

considerable for a smaller specimen of the same thickness. The effect would 

be to lower the buckling stress. The formulas of this report do not consider 

the effect of shearing deformations, 

Conclusions 

The buckling stress of curved plywood plates in shear can be computed by adding the buckling stress in torsion of a complete cylinder of the same length and 

curvature as the curved plate to the buckling stress of a flat plate, provided 

the curved plate is not half or nearly half of a cylindrical shell. 

The buckling stress of a curved plate of sandwich construction can be computed in the same manner, provided in addition, that the plate is of such dimensions that the effects of transverse shearing deformations are negligible. 

Rept. No. 1571 -7-
APPENDIX I 

3 Cylinder of Sandwich Construction in Torsion 

The derivation of formula (2), from the analysis of Forest 

Products Laboratory Report No. 1529 is outlined in this appendix. The notation of Report No. 1529 is used throughout. 

It is assumed that the facings are isotropic and that the Young’s modulus and 

the shear modulus of the core can be neglected in calculating the necessary 

constants for determining the behavior of the sandwich. Then 

As in Forest products Laboratory Report No, 1558, page 18, it follows that 

Equations (1 1) of Forest Products Laboratory Report No. 1529 then become 

and equations (16) become 

In like manner equations (19) become 

3

The analysis presented here was made by H. W. March. 

Rept. No. 1571 -8-
Therefore, equations (19) reduce to the form that they would have for an isotropic cylinder with a Young’s modulus Ea. 

It follows that the expression for the strain energy W1 is identical with that 

for an isotropic cylinder with a Young’s modulus Ea and a shear modulus 

when is the value of Poisson’s ratio in the facings. 

In like manner the constants in equation (22) can be replaced by the constants 

appropriate to the bending of a sandwich with isotropic facings, with the contribution of the core being neglected. Consequently the constants b1, b2, etc. , 

that appear in equation (25) have the following values: 

b1 = b2 = E1 b4 = b6 = 0 

b3 + b5 = 2E1 

Hence, equation (25) will have the same form as that for an isotropic cylinder 

having a Young’s modulus E1. 

For a cylinder of isotropic material with Young’s modulus E, equation (29) of 

Report No. 1529 (See equations (30), (31), and (32) of that report and note that 

g = 0) can be written in the form 

where 

f

li and f2i are the values of f1 and f2, as defined by equation (32) for an isotropic material. In this expression the first term comes from the first term 

in brackets in (29) together with its coefficient, while the second term comes 

from the second term in brackets in (29) together with its 

while the conto 

coefficient. Now, 

for isotropic material, the constants cij are proportional to 

stants M, P, S, Q, and N are either zero or proportional Hence, 

the whole first bracket expression is proportional to E. The factor E also 

Rept. No. 1571 -9-
appears in the denominator of the expression for fli. Let E fli = In the 

the second bracket in (29) the coefficients b1, b2, b3, b4, b5, b6 are either 

zero or proportional to E, and, consequently the whole bracket expression is 

proportional to E1. Then 

can be written for an isotropic cylinder where 

tional to E. 

For a sandwich cylinder the first term is to be that: for an isotropic cylinder 

with E replaced by Ea, and the second term is to be that for an isotropic 

cylinder with E replaced by E1. 

Hence, for a sandwich cylinder 

where fli and f2i are to be calculated from equations (32) for an isotropic 

cylinder with Young’s modulus E. The appearance of this modulus is purely 

incidental, since it is found in both the numerators and the denominators of 

the expressions for f1i and f2i and cancels out. 

In the expression for 

let 

Then 

Rept. No. 1571 -10-
where 

This is the value of k of Forest Products Laboratory Report No. 1529 for an 

isotropic cylinder for which 

APPENDIX 2 

Testing of Plywood Coupons 

Bending Tests . 

To obtain the bending moduli of elasticity (E1 and E2) of the plywood, 1- by 

6-inch strips having their face grain direction parallel to their length were 

tested on a 4-inch span and 1- by 4-1/2-inch specimens having their face grain 

direction perpendicular to their length were tested on a 2-1/2-inch span. The 

load was applied at the center of the span by means of the movable head of a 

mechanical testing machine. The load was measured on a scale that could be 

read to the nearest gram. Deflections at the center of the span were measured 

with a dial reading to 0.001 inch. The main spring had been removed from the 

dial so that the load exerted by the dial stem was negligible. 

Tension Tests 

Tension tests were made on 1-inch strips to obtain the moduli of elasticity; 

for example, Ea (face grain direction parallel to direction of loading), Eb (face 

grain direction perpendicular to the direction of loading), and E45. (face grain 

direction at an angle of 45° to the direction of loading). The specimens were 

11 inches long. The ends were held in Templin Grips fastened to the testing 

machine by bolts passed through spherical bearings. The strain over a 2-inch 

gage length was measured with an extensometer reading to 0.0001 of an inch. 

Rept. No. 1571 -11-
-- 

Computation of Results 

Values of EL were computed by the equation 

where = 0.045 for yellow birch plywood. 

Values of the shearing modulus, μLT' were computed by the equation 

where = 0.046 for yellow birch plywood. 

This formula was obtained from the equation for given in Forest 

Products Laboratory Report No. 1317. 

APPENDIX 3 

Sample Computations of Buckling Stresses 

Sandwich Construction 

Specimen data 

Curvature r = 93 inches 

Width a = 50 inches 

Length b = 50 inches 

Total thickness h = 0.149 inch 

Facing thickness f = 0.012 inch (24 ST alclad aluminum) 

Core thickness c = 0.125 inch (end-grain balsa) 

Modulus of elasticity 

of facing Ef = 10 x 106 p.s.i. 

Rept. No. 1571 -12-
-- 

Then 

From curve of figure 6, k' = 0.245. 

Therefore, 

for determining ks 

from figure 8 of Report No. 1316 ca = 22.9. 

Then 

Therefore, 

and the buckling stress of the curved plate = 

Plywood Construction 

Specimen data 

Curvature r = 8 inches 

Width a = 10.8 inches 

Length b = 20 inches 

Total thickness h = 0.059 inch 

Plywood is three-ply 1: 2: 1 yellow birch = 0° (face grain parallel to cylinder 

axis). 

Elastic properties (from coupon tests) 

E1 = 2.32 x 10 6 p. s. i. 

E2 = 0.435 x 10 6 p. s. i. 

Rept. No. 1571 -13-
Ea = Eb = 1.342 x 10 6 p. s. i. 

EL = 2.602 x 10 6 p. s. i. 

μ

LT = 2.01 x 10 5 

p. s. i. 

Then 

Then entering the curves of figure 2 and interpolating to the value of 0.077 for 

Therefore, 

For determining 

Enter curve of figure 130, Forest Products Laboratory Report No. 1316-HI 

and obtain 

Enter curve of figure 129 of Report No. 1316-HI and obtain 

Compute 

Enter curve of figure 12 of Report No. 1316, and obtain 

Therefore, 

and the buckling stress of the curved plate = 

4 It should be noted that of this report refers to the angle of the face grain to 

the cylindrical axis, but of Reports 1316 and 1316-H refers to the angle 

between the short side of the specimen and the face grain direction. 

Rept. No. 1571 -14- .5-26 

Figure 1.--Bucklingof curved plate of sandwich construction in shear. 

Z M 73407 F 





Figure 7.--Sketch of curved-plate specimen. 

Z M 78984 F 
Figure 8..--Experimental buckling stress of a curved plate in 

shear vs. theoretical buckling stress of a complete cylinder

in torsion. 

Z M 78886 F 
Figure 9.--Experimental vs. theoretical buckling stresses of 

curved plates in shear. 

Z M 78886 F 
FPL FILING SYSTEM DESIGNATION-4-5-5 

Kuenzi, Edward William 

to shear. 3rd ed. Madison, Wis., U.S. 

Forest Products Laboratory, 1962. 

Stability of a few curved panels subjected 

14 p., illus. (F.P.L. rpt. no. 1571) 

Presents results of an experimental study on 

the buckling of curved plywood panels and one 

large sandwich panel. 

Kuenzi, Edward William 

Stability of a few curved panels subjected 

to shear. 3rd ed. Madison, Wis., U.S. 

Forest Products Laboratory, 1962, 

14 p., illus. (F.P.L. rpt. no. 1571) 

Presents results of an experimental study on 

the buckling of curved plywood panels and one 

large sandwich panel. 

Kuenzi, Edward William 

to shear. 3rd ed. Madison, Wis., U.S. 

Forest Products Laboratory, 1962. 

Stability of a few curved panels subjected 

14 p., illus. (F.P.L. rpt. no. 1571) 

Presents results of an experimental study on 

the buckling of curved plywood panels and one 

large sandwich panel. 

Kuenzi , Edward William 

to shear. 3rd ed. Madison, Wis., U.S. 

Forest Products Laboratory, 1962. 

14 p., illus. (F.P.L. rpt. no. 1571) 

Stability of a few curved panels subjected 

Presents results of an experimental study on 

the buckling of curved plywood panels and one 

large sandwich panel.